Abstract: This is the second paper in our completeness series which addresses some ofthe issues raised in the previous article by Johnston, Teodoro and Hendry 2007in which we developed statistical tests for assessing the completeness inapparent magnitude of magnitude-redshift surveys defined by two flux limits.The statistics, Tc and Tv, associated with these tests are non-parametric anddefined in terms of the observed cumulative distribution function of sources;they represent powerful tools for identifying the true flux limit and-orcharacterising systematic errors in magnitude-redshift data. In this paper wepresent a new approach to constructing these estimators that resembles an-adaptive smoothing- procedure - i.e. by seeking to maintain the same amountthe information, as measured by the signal-to-noise ratio, allocated to eachgalaxy. For consistency with our previous work, we apply our improvedestimators to the Millennium Galaxy Catalogue MGC and the Two Degree FieldGalaxy Redshift Survey 2dFGRS data, and demonstrate that one needs to use asignal-to-noise appropriately tailored for each individual catalogue tooptimise the performance of the completeness estimators. Furthermore, unlesssuch an adaptive procedure is employed, the assessment of completeness mayresult in a spurious outcome if one uses other estimators present in theliterature which have not been designed taking into account -shot noise- due tosampling.